Optimization of Motor Rotor: Slot Shape, Frame Size, Rotor Diameter


Designing Squirrel Cage Rotor Slots with High Conductivity


There has been, in recent years, an effort to make cast copper rotors for industrial use induction motors. The objective is to make motors more efficient because of the higher conductivity of copper. In addition, the reduced losses in such motors may lead to better design flexibility and therefore motors that are more compact.

In this paper we examine the tradeoff between running efficiency and starting performance. In order to understand how induction motors work it is necessary to have a good model for the conduction properties of the conductors in the slots of the rotor. We employ a simple model that has been developed for this purpose. We propose to examine the frequency response of the rotor bar impedance for indications of how the motor will work, taking into account not only running efficiency and starting performance but also stray load losses which are also affected by rotor bar impedance.


For illustration, Fig. 1 shows a comparison of torque vs. speed for two identical notional induction motors, differing only in the value of rotor resistance, commonly noted as R2. The characteristic curve with higher resistance would, for all load torques, operate at a somewhat lower running speed (and hence higher slip and lower efficiency) than the machine with the lower resistance. On the other hand, starting torque is lower for the low resistance rotor and this might lead to unsatisfactory performance in some applications.

The use of copper as the conductor of induction motor rotors would typically lead to improvements in efficiency, relative to motors using aluminum. This is often done in large motors in which squirrel cages are fabricated from bars of material brazed to end rings. In such machines it is common to employ 'starting' bars of higher resistivity material. Our objective here, however, is to discuss ways of designing rotor slots for fabrication by casting a single material, as is commonly done with aluminum as the rotor conductor material.

Figure 1: Notional Torque Speed Curves

In some induction motors, what is called the 'deep bar' effect, or distribution of rotor currents due to eddy currents, is taken advantage of. When the rotor is stationary or turning slowly the frequency of rotor currents is relatively high, the currents crowd to the top of the rotor bar and the resistive part of impedance is relatively high.

Figure 2: Comparison of Slot Shapes

Shown in Fig. 2 are three different possible slot shapes. On the left is the shape used in the original cast aluminum rotor. It has a characteristic tapered shape so that the teeth between rotor bars are of uniform section. It tapers toward the rotor surface.

Fig. 3 shows the real and reactive parts of the impedance of the slot designed for aluminum with two different materials: aluminum and copper.

Figure 3: Aluminum and Copper Slot Resistance

The frequency range shown in Fig. 3 encompasses not only starting and running frequencies, but also the frequencies of currents in the rotor bars due to the various space harmonics of the stator winding. As can be seen in the figure, the copper rotor has substantially lower resistance over the whole frequency range, but the difference is largest at running conditions. At higher frequencies the smaller skin depth of copper causes the impedance of those bars to increase.

Now the question at hand is this: Can we shape the rotor bars to take advantage of the higher conductivity of copper to produce good running efficiency and yet have satisfactory starting performance? We do not believe we have found the optimum bar shape yet, but we can at least illustrate the question with a comparison of the other two bar shapes shown in Fig. 2. Fig. 4 shows a comparison of the slot resistance of those two bars over the same frequency range.

The first copper bar was designed with a narrow top within which the relatively high frequency currents of starting flow, producing a higher resistance at start. Note that this is only partially successful, as the second bar design, B actually results in larger resistance at starting. This is because the narrow slot between the top, starting bar and the rest of the conductor is effective at isolating starting frequency currents to the starting bar. At the same time, note that Bar A results in higher resistance at harmonic frequencies, indicating possibly higher stray load losses.

Figure 4: Comparison of two copper rotors (Slot A and Slot B of Fig. 2)


Induction motors are represented for the purpose of design analysis by an equivalent circuit as shown in Fig. 5. Some components of the rotor leakages and resistances of each section are frequency dependent and therefore functions of rotor frequency and so speed.

Figure 5. 'Alger' Equivalent Circuit Model

To establish rotor resistances and reactances we use a simple technique for representing the rotor bar as a ladder network. The bar is divided into a relatively large number of slices, oriented in a direction perpendicular to the rotor surface. Each such slice is represented by a simple 'tee' circuit consisting of inductance, representing cross-slot flux and resistance, representing the current carried by that part of the slot.

Figure 6: Element Model

To find the values of the slot parameters make reference to Figure 7.

Figure 7. Illustration of calculation of slot parameters

Note that, from the Ampere's Law contour shown, magnetic field crossing the slot at some vertical position y is related to all current in the slot below that position. From the Faraday's Law contour we see that all of the inductance elements in the slot are in series and the electric field (voltage per unit length) driving current is the sum of voltages across all inductive elements below the position y. Then if we represent the slot as a series connection of basic 'tee' elements, the inductance per unit length of a section is:

where d is the height of the section and w is width of the slot at that position. Resistance per unit length is: where σ is material conductivity.

The tee elements are accordingly connected together as is shown in Fig. 8.

Figure 8: Ladder representation of slot impedance

A more sophisticated technique might have been used. A good example of this is the elegant formulation of Williamson using a finite element model for each slot embedded in a circuit model of the machine as shown in Fi.g 5 . In this case it was felt that the accuracy achieved by the ladder network would probably be sufficient, and the resulting efficiency of calculation makes it possible to draw some conclusions regarding optimization of the machine.


Many different types of slot shapes are possible in induction motors. In large machines it is typical to use slot shapes that both concentrate currents near the rotor surface and can withstand the heating associated. It is also necessary to have enough conductor area near the surface to produce relatively low loss at the belt and zigzag harmonic frequencies. For small motors we have had some success with the slot shape shown in Figure 9.

Figure 9: Modified Kite Form Bar Shape
Figure 10: Slot Geometry

Larger machines may require different bar forms. Figure 10 shows a geometry that has a discrete 'starting bar' to carry high frequency currents at starting conditions plus a 'leakage slot' to magnetically isolate the starting bar from the main conductor under starting conditions. This type of bar is shown in Figure 10.The 'main' part of the slot is a trapezoid that is narrower at the bottom than at the top to allow for constant width of teeth between slots. The top and bottom part of the slot is bounded by semicircles of appropriate radius. The 'leakage slot has radial height h and the starting bar has radius Rb. If we define the slot factor to be the ratio of slot width to slot pitch (slot plus tooth) at the top of the trapezoidal (main) part of the slot to be:


To illustrate the impact of using copper in a machine we adopt a 5.5 kW, 50 Hz machine built for pump application so that starting torque is an important feature. The rotor slot we assume here is similar to the slot in the actual machine (in which the starting bar is not exactly cylindrical as we assume. The bar is defined by a starting bar diameter of 4.2 mm, a leakage slot that is 3 mm high and 1 mm wide, 28 rotor bars in a rotor that is 110 mm in diameter and a ratio of inner to maximum slot widths of . The slot as constructed is shown in Fig. 11. The horizontal lines represent section widths.

Figure 11: Example Slot Shape

Using standard analysis techniques and the slot impedance estimation technique described here, we can estimate machine performance for cast aluminum and cast copper. The analysis technique has been compared with the actual 5.5 kW motor and the results are reasonably accurate for both copper and aluminum rotor cages. The frequency response for the two conductors is shown in Fig. 12 and the Torque-speed curves are shown in Fig. 13.

Figure 12. Frequency Response Comparison: Aluminum Vs. Copper
Figure 13: Comparison of copper vs. aluminum in 5.5 kW motor

In this machine the copper rotor achieves better performance by taking advantage of the deep bar effect to maintain starting torque while still improving efficiency (89.8% vs 87% for Al).


In an attempt to understand how variations in slot geometry might affect machine performance a short parametric trade was carried out by varying the starting bar diameter from 2 to 10 mm, holding the leakage slot geometry constant, and then varying the leakage slot height from 1 to 5 mm while holding the starting bar diameter at 4.2 mm. The resulting extremes of slot geometry are shown in Fig. 14 and 15.

Figure 14: Slot Geometry Varying Starting Bar

Machine performance has been estimated for the ranges of slot geometries shown in Fig. 14 and 15, assuming copper to be the cage material. Selected performance parameters are shown in Fig. 16 and 17. Fig. 16 shows variation of efficiency and starting torque over variation of starting bar radius, holding the other defining parameters constant.

Figure 15: Slot Geometry Varying Leakage Slot Height
Figure 16: Parametric Study vs. Starting Bar Diameter
Figure 17: Parametric Study vs. Leakage Slot Height

In the case of leakage bar height there is a tradeoff: increasing the leakage height consistently increases starting torque while also consistently reducing efficiency. This is to be expected as increasing the rotor leakage height pushes the main part of the slot down and reduces available slot width, leaving less conductor in running conditions and increasing running resistance. On the other hand, increasing the starting bar diameter seems to consistently increase efficiency as it increases total conductor available. There is, however, a maximum starting torque at a bar diameter very close to the diameter of the example machine.


This brief study is an attempt to understand how to make appropriate use of different conducting materials in the squirrel cages of cast rotors. In particular, we were interested in how to properly use the higher conductivity of copper in such motors.

As should be clear, the higher conductivity of copper increases skin effect and therefore makes deep bar and multiple cage effects more pronounced. This can be seen from the frequency response plots that examine the impedance per unit length of the rotor slots. High frequency impedance of the rotor slots has an impact on stray load and no-load losses. Medium frequency (around line frequency) impedance effects starting performance. Low frequency impedance (about slip frequency) affects running efficiency. While this has been known for some time, the frequency response plots shown here allow for visualization of these effects. They can tell why a narrow conductor filled slot above the main bar is probably not as effective as a starting bar with a narrower leakage bar below.

We have also confirmed the adequacy of calculating rotor cage impedances using a simple flux tube (one dimensional finite element) model for simple trade studies.

  1. Modeling of Rotor bars with Skin Effect for Dynamic Simulation of Induction Machines, J. Langheim, Conference Record, IEEE Industry Applications Society Annual Meeting, 1989, pp 38-44.
  2. Induction Machines, Philip Alger, Gordon and Breach, 1970. This is actually an extention to Alger's model, explained in Electric Motor Handbook, H. Wayne Beaty et al. McGraw Hill, 1998
  3. Calculation of cage induction motor equivalent circuit parameters using finite elements, S. Williamson, M.J. Robinson, Proc. IEE (Part B) Vol 138, No. 5 pp 264-276, (1991)


SEW-Eurodrive, a large manufacturer of industrial drive systems, has made the CRM an integral part of a broad range of drive motors upgraded to European efficiency classification EFF1 (50 Hz) and EPAct (60 Hz) efficiencies. Their studies indicated that in many instances, the efficiency increase permitted use of the same frame size with a copper rotor as the current production aluminum rotor (Brush 2002, Kimmich 2005). Design studies showed that the EFF1 aluminum rotor motor would require moving up one frame size and would require increasing the size of the entire motor/gear package. This would be expensive and had the additional disadvantage of making retrofitting with a more efficient motor impossible. In the redesign of the motor lines to EFF1, direct substitution of copper for aluminum was found to meet the objectives for the smaller motors while the larger motors were entirely redesigned to best utilize the copper. The copper redesigns maintained the outside motor dimensions and achieved the desired torque at various points on the torque-load curve and controlled the in-rush currents in the starting region.

Table 1 presents efficiency data for four SEW aluminum and copper rotor motors at 60 Hz. The 1.5 Hp motors essentially have the same layout of stator and rotor laminations; i.e. the aluminum rotor bars have simply been replaced by the die-cast copper. But the lamination steel has been upgraded from material with losses of 8 W/kg to a better grade with losses rated at 4 W/kg. In contrast, the high efficiency larger motors have a completely new lamination and stator design. The design modifications relate to the starting behavior discussed below. The data in Table 1 show that the copper rotor leads to a significant increase in efficiency.

In taking the decision to use copper in the rotor for this series of industrial drive motors to reach EFF1 minimum efficiencies, SEW conducted an extensive modeling study comparing the size, weight and overall costs of motors of equivalent efficiency using aluminum in the rotor. The finding was that, for the motors discussed here, the use of copper in the rotor cage allowed reductions in rotor diameter, in iron required for laminations and in stator copper windings. The copper rotor motors are one frame size smaller than the the aluminum rotor design would have allowed. Overall there was an accompanying reduction in total manufacturing costs; the cost of the motor with an aluminum rotor at a given EFF1 efficiency ranged from similar to 15% higher than the copper version. In these examples, weight savings of up to 18% and cost savings of up to 15% were effected. This cost saving for the copper rotor motor was in spite of the die-casting component of the copper rotor being typically three times more costly than the aluminum rotor.

Analysis by U.S. manufacturers of 7.5 and 15 Hp motors and assembled by CDA as a composite equivalent U.S. motor meeting EPAct efficiency standards came to similar conclusions. The die-cast copper rotor motors would be 18 to 20% lighter and 14 to 18% less expensive to build than the aluminum rotor motor at the same efficiency when a frame size reduction was possible. When a frame size reduction was not possible, reductions and weight and cost were still indicated in the design studies, but the percentage reductions were in the single digits for the copper rotor machine.